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applicazioni alla logica

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\maketitle
\section{Introduction}
\section{Freyd cover of a category}
\setcounter{chapter}{-1}
\chapter{Introduction}
\chapter{Freyd Cover}
\section{Freyd cover of a category}
\begin{definition}[Comma Category]
Let $\mathcal{A,B,C}$ locally small categories and let $F: \mathcal{A} \to \mathcal{C}, G: \mathcal{B} \to \mathcal{C}$ two functors
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Now, commutativity at the level of $Set$ is ensured by universal property of $Y^X$ in $Set$ plus the coerence condition of $(Y^X)_{a \leq b}$ with respect to $a \to b$. Now, in $\mathcal{H}$ we have that $c \land a \leq b$ we have that $c\land a \leq b \iff c \leq a \to b$ so we have commutativity at the level of $\mathcal{H}$. It remains to check the well definition of the abstraction: let $Z \neq \emptyset$, then $h(m) \in \mathcal{H}(1_\mathcal{H}, c)$ for $m \in Z$, hence $1_\mathcal{H} \leq c$ and so, since $c \land a \leq b$, then $a \leq b$ and so $\hat{u} (m) \in (Y^X)_{a \leq b}$ wihch ensure well definition of abstraction arrows. Notice that uniqueness of abstraction follows directly form uniqueness of abstractions in $Set$ and $\mathcal{H}$. Thus $Cov(\mathcal{H})$ is a bicartesian cloed category.\\
Concerning $\Pi_2$, the proof is identical to \ref{completenesscomma} since the $\mathcal{H}$ level of the exponential object is the exponential object in $\mathcal{H}$ and so $\Pi_2$ is a bicartesian closed functor and we are done.
\end{proof}
\begin{remark}[Structure of $Cov(\mathcal{C})(1, A \oplus B)$] \label{global section in cover rmk}
Let $(u,v): 1 \to (X \oplus Y, a \lor b, f \oplus g)$ and consider $u: 1_{Set} \to X \oplus Y$. Since the coproudct in $Set$ is precisely the disjoint union, then $u(\star)$ is either in $X$ or in $Y$. Assume $u(\star) = x_0 = \iota_X(x_0) \in X$ for some $x_0 \in X$, then $(f \oplus g) \circ u = \Gamma v \circ un $ implies $v(\star)= (\iota_1)_\mathcal{C} \circ f (x_0)$. Since this argument is simmetrical if $u(\star) \in Y$ we have the following decomposition:
\[
Cov(\mathcal{C})(1, A \oplus B) \cong Cov(\mathcal{C})(1, A ) \sqcup Cov(\mathcal{C})(1, B)
\]
\end{remark}
\begin{definition}
Let $\mathcal{H}$ be an Heyting Algebra, then we call its Feryd cover the Frey-Heyting cover of $\mathcal{H}$
Let $\mathcal{H}$ be an Heyting Algebra, then we call its Freyd cover the Freyd-Heyting cover of $\mathcal{H}$
\end{definition}
\begin{definition}
We say that $\textbf{biCC}$ is the category with objects bicartesian closed categories and morphisms functors preserving the bicartesian structure up to isomorphism
@@ -745,16 +750,55 @@
\arrow[""{name=1, anchor=center, inner sep=0}, "{(h,Fk)}", from=2-3, to=3-3]
\arrow[between={0.2}{0.8}, maps to, from=0, to=1]
\end{tikzcd}\]
where $F\cdot f: X \to \Gamma (FA)$ is the set function defined sending $x \in X$ into the global setion of $FA$ given by $F( f(x)) : 1_\mathcal{B} \to FA$
where $F\cdot f: X \to \Gamma (FA)$ is the set function defined sending $x \in X$ into the global setion of $FA$ given by $F( f(x)) : 1_\mathcal{B} \to FA$
\end{proposition}
\begin{proof}
By lemma \ref{cover of cc is cc} $Cov(-)$ exentend to a class function from \textbf{biCC} to itself. Then notice that the action of $Cov(-)$ over arrows of \textbf{biCC} is well defined since the commutativity condition of the arrow is preserved under the action of $F$. Indeed by definition of $F \cdot f$ and $F \cdot g$ we have:
\[
(F \cdot g) \circ h = F(\Gamma k) \circ (F \cdot f)
\]
Ando so for $F: A \to B$ we proved $Cov(F): Cov(\mathcal{A}) \to Cov(\mathcal{B})$ is a well defined functor. Moreover, clearly $Cov(Id_\mathcal{A})= Id_{Cov(\mathcal{A})}$ and by definition of $F \cdot f$ thaking $F: \mathcal{A} \to \mathcal{B}$ and $G: \mathcal{B} \to \mathcal{C}$ we have that $Cov(G \cdot F) = Cov(G) \cdot Cov(F)$. Thus $Cov(-): \textbf{biCC} \to \textbf{biCC}$ is a well defined endofunctor
\end{proof}
\end{proposition}
\begin{corollary}
Taking posettal reflection of a lattice - i.e. an ordered bicartesian closed category - gives rise to an endofunctor:
\[
Pcov(-): \textbf{bLat} \to \textbf{bLat}
\]
where $Pcov(-) := Pos \circ Cov (-) : \textbf{bLat} \to\textbf{bLat}$ and \textbf{bLat} is the category of bounded lattices - i.e. lattice with top and bottom objects - with lattice homomorpishms as arrows.
\end{corollary}
\begin{proof}
By lemma \ref{cover of bicartesian} the cover of a lattice has a bicartesian structure and by lemma \ref{cover of cc is cc} the cover of a lattice is also a cartesian closed category. Therefore we are done, since the posettal reflection of a bicartesian closed category is a bounded lattice (the posettal reflection preserves bicartesian closed struture of the category)
\end{proof}
\section{Application to Logic}
After defining the construction of Freyd Cover and proving some baisc property of this constuction we are ready to use the categorical property of $Cov(-)$ to show some logical poperty of intuitionistic logic. In this section we will prove categorically that the propositional logic satisfy the disjunction property using the Freyd Cover construction.
\begin{theorem}
Let the following disjunction $\phi \lor \psi$ be derivable in the intuitionistic propositional logic. Then either $\phi$ or $\psi$ is derivable.
\end{theorem}
\begin{proof} \, \\
\textbf{Idea}: We will apply our construction to the Lindembaum-Tarski algebra of the initial intuitionistic propositional logic $\mathcal{T}_0$. Notice that the projection $\Pi_2: Cov(\mathcal{A}_i(\mathcal{T}_0)) \to \mathcal{A}_i(\mathcal{T}_0)$ "preserves" the Lindembaum component of element of $Cov(\mathcal{A}_i(\mathcal{T}_0))$,therefore a clever model build upon $Cov(\mathcal{A}_i(\mathcal{T}_0))$ respects the categorical semantics of $\mathcal{T}_0$. We will use this property to construct, from the fact that $\phi \lor \psi$ holds, a global section of $\phi$ or $\psi$, proving the disjunction property. \\
\textbf{Proof}: Let $\mathcal{T}_0$ be the initial intuitionistic propositional theory and consider the Lindembaum-Tarki algebra $\mathcal{A}_i(\mathcal{T}_0)$. Notice that $Pcov(\mathcal{A}_i(\mathcal{T}_0))$is an heyting algebra, so we can define a model
\[
\nu : Frm(\mathcal{T}_0) \to Pcov(\mathcal{A}_i(\mathcal{T}_0))
\]
in the following way: let $A$ be a propositional variable, then
\[
A \to \nu(A)= [(X_A, [A], e_A)]
\]
where $X_A$ is the terminal object in $Set$ if $1 \leq_{\mathcal{A}_i(\mathcal{T}_0)} [A]$ and initial otherwise:
\[
X_A := \{ x = 0 | 1 \leq_{\mathcal{A}_i(\mathcal{T}_0)} [A] \}
\]
notice that $[(X_A,[A],e_A)]$ is a well defined element of $Pcov(\mathcal{A}_i(\mathcal{T}_0))$. In fact if $A$ is the top in $\mathcal{A}_i(\mathcal{T}_0)$ then the set global section of the class $[A]$ is terminal and so $e_A$ is uniquely defined. Otherwise $X_A$ is inital in $Set$ and thus again $e_A$ is uniquely defined. Now, this definition can be extended inductively by lifting the Lindembaum-Tarski evalutation to the Freyd Cover, namely by setting $\nu(\psi \land psi) = \nu (\psi) \times \nu (\psi)$, $\nu (\psi \lor \psi) = \nu(\phi) \oplus \nu(\psi)$ and $\nu(\phi \to \psi)= \nu(\psi)^{\nu(\phi)}$. Therefore for an arbitrary formula $\phi$ we have:
\[
\nu(\phi) = [(X_\phi, [\phi], e_\phi)]
\]
for some $X_\phi$ in $Set$ and $e_\phi : X_\phi \to \Gamma [\phi]$. Notice that in this model the evaluation of a propositional variable $A$ and $\neg A$ is neither the top or the bottom (this follows immediatly by construction). \\
Let $\phi \lor \psi$ be a tautology in $\mathcal{T}_0$, therefore \[\nu(\phi \lor \psi) = 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} = [(1_{Set}, 1_{\mathcal{A}_i(\mathcal{T}_0)},un)]
\]
Now, since $\nu (\phi \lor \psi)= \nu(\phi) \oplus \nu(\psi)$ there exists a global section $g: 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} \to [(X_\phi , [\phi] , e_\phi)] \oplus [(X_\psi , [\psi], e_\psi)]$.
By remark \ref{global section in cover rmk} $g = (g_1,g_2): 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} \to \nu(\phi)$ or $g = (g_1,g_2): 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} \to \nu(\psi)$. Now, in the first case $e_\phi (g_1(\star)) \in \Gamma [\phi]$ hence $\phi$ is provable in the logic. In the second case $e_\psi (g_1(\star)) \in \Gamma [\psi]$ and so $\psi$ is provable in the logic. Thus the disjunctin property is proved and so we are done.
\end{proof}
\printbibliography
\end{document}